An extension of ergodic theory for Gauss-type maps
The impetus to this work is the need to show that for positive reals and , the functions
span a weak-star dense subspace of if and only if . Here, is the subspace of which consists of those functions whose Poisson extensions to the upper half-plane are holomorphic. In earlier work in the context, we showed the relevance of the analysis of the dynamics of the Gauss-type mapping mod for this problem (if , which can be assumed by a scaling argument). For , the ergodic properties of the absolutely continuous invariant measure on the interval turned out to be crucial. In the present setting, although the norm in is the same as in , in the real sense, it is much finer. The corresponding real space is , which consists of all the functions in whose modified Hilbert transform is also in . From the real perspective, our task is clearer: We need to show that the functions
span a weak-star dense subspace in precisely when . The predual of is identified with a space of distributions on , obtained as the sum of and , where is the codimension subspace of of functions with integral . While the space consists of distributions, it also can be said to consist of weak- functions, and a theorem of Kolmogorov guarantees that the viewpoints are equivalent. It is in a sense the extension of which is analogous to having BMO as the extension of . Whereas transfer (and subtransfer) operators usually act on of an interval (or, more generally, on the finite Borel measures on that interval), here we consider the corresponding operators acting on the restriction of to the interval in question, denoted . In the convex body of invariant absolutely continuous probability measures an element is ergodic if it is an extreme point. In the setting of infinite ergodic theory, which is more relevant here, ergodicity means that no element of on the interval can be invariant (under the transformation, or, which is the same, under the transfer operator). We study mainly a particular instance of infinite ergodic theory, and extend the concept of ergodicity by showing that for the transformation mod on , (i) for , there is no nontrivial subtransfer operator invariant distribution in , whereas (ii) for , there is no nontrivial transfer operator invariant odd distribution in . The oddness helps in the proof, but we expect it to be superfluous. The conclusion is nevertheless strong enough to supply an affirmative answer to our original density problem. To obtain the results (i)-(ii), we develop new tools, which offer a novel amalgam of ideas from Ergodic Theory with ideas from Harmonic Analysis. We need to handle in a subtle way series of powers of transfer operators, a rather intractable problem where even the recent advances by Melbourne and Terhesiu do not apply. More specifically, our approach involves a splitting of the Hilbert kernel induced by the transfer operator. The careful analysis of this splitting involves detours to the Hurwitz zeta function as well as to the theory of totally positive matrices.
Key words and phrases:Transfer operator, Hilbert transform, completeness, Klein-Gordon equation
2000 Mathematics Subject Classification:Primary 42B10, 42B20, 35L10, 42B37, 42A64. Secondary 37A45, 43A15.
1.1. An elementary example: the doubling map of an interval
Let us consider the doubling map of the unit interval , given by mod ; to be more precise, we put on , and on . For and , we have the identity
where is the associated transfer operator
The function (and the corresponding absolutely continuous measure ) is said to be invariant with respect to the doubling map if . We quickly check that the constant function is invariant, and wonder if there are any other invariant functions beyond the scalar multiples of . To analyze the situation, Fourier analysis comes very handy. We expand the function in a Fourier series
which actually need not converge pointwise, but this does not bother us. The Fourier series associated with is then
and, by iteration,
If solves the more general eigenvalue problem for some complex nonzero scalar , then we see by equating Fourier coefficients that we must have
for . By plugging in , we derive from the above equation (1.1.1) that is the only possibility, provided that . Moreover, for , we know from the Riemann-Lebesgue lemma that
which lets us to conclude from (1.1.1) that
provided that . In this case, is of course constant, and if the constant is nonzero, then we also know that . In particular, the only invariant functions in are the constants. This observation is an equivalent reformulation of the well-known ergodicity of the doubling map with respect to the uniform measure on the interval (see below).
Observation: As we look back at the argument just presented, we realize that we did not use all that much about the function , just that the conclusion of the Riemann-Lebesgue lemma holds. So in principle, we could replace by a finite Borel measure, and obtain the same conclusion, if the Fourier coefficients of the measure tend to at infinity. Such measures are known as Rajchman measures, and have been studied in depth in harmonic analysis. But the point of view we want to present here goes beyond that setting. We are in fact at liberty to replace by a distribution with a periodic extension, so that it has a Fourier series expansion, and so long as its Fourier coefficients tend to as , the argument works, and tells us that the constants are the unique -invariant elements of this much wider space of distributions. Such periodic distributions which Fourier coefficients which tend to as deserve to be called Rajchman pseudomeasures (cf. ). This uniqueness within the Rajchman pseudomeasures can be understood as an extension of standard ergodic theory for the doubling map with respect to the constant density . Indeed, an easy argument shows that the following are equivalent, for an invariant probability measure :
(i) is ergodic, and
(ii) whenever is a finite (signed) invariant measure, absolutely continuous with respect to , then is a scalar multiple of .
This is probably well-known. For completeness, we supply the relevant argument. Note first that we may restrict to real measures and real scalars in (ii). The implication (i)(ii) is pretty standard and runs as follows. By replacing by the sum of and an appropriate scalar multiple of , we reduce to the case when has signed mass . Then, unless , we split into positive and negative parts, which are seen to be left invariant by the transfer operator, as otherwise the transfer operator applied to would have smaller total variation than itself. But then the support (or rather, carrier) sets for the positive and negative parts are necessarily invariant under the transformation, in violation of the ergodic assumption (i), and the only remaining alternative is that , i.e., the assertion (ii) holds. The remaining implication (ii)(i) is even simpler. We prove the contrapositive implication, and assume that (i) fails, so that is not ergodic. Then is not an extreme point in the convex body of all invariant probability measures, and hence it splits as a nontrivial convex combination of two invariant probability measures. Both measures are assumed different than itself, and each one is obviously absolutely continuous with respect to , which shows that (ii) fails as well.
1.2. The Gauss-type maps on the symmetric unit interval
It was the fact that the doubling map is piecewise affine that made it amenable to methods from Fourier analysis. This is not the case for the Gauss-type map acting on the symmetric interval , defined in the following fashion. First, we let denote the even-fractional part of , by which we mean the unique number in the half-open interval with . The Gauss-type map is given by the expression
Here and in the sequel, is assumed real with . The basic properties of are well-known, see, e.g. . We outline the basic aspects below, which are mainly based on the work of Thaler  and Lin . For , the set acts as an attractor for the iterates under , and inside the attractor , the orbits form -cycles. Here, denotes the symmetric interval . For , on the other hand, we are in the setting of infinite ergodic theory, where is the ergodic invariant measure. The reason is that the endpoint (which for all essential purposes may be identified with the left-hand endpoint for the dynamics) is only weakly repelling. The tranfer operator linked with the map is the operator which can be understood as taking the unit point mass at a point to the unit point mass at the point . To be more definitive, for a function , we write as an integral of point masses,
understood in the sense of distribution theory, and say that
which is seen to be the same as the more explicit formula
which has the added advantage that the values off the interval are declared to vanish. The behavior of is rather uninteresting on the attractor , and for this reason, we introduce the subtransfer operator which discardsthe point masses from the attractor. In other words, we put
In more direct terms, this is the same as
which we see from (1.2.3). Here, , as before. For , the -orbit of a point falls into the attractor almost surely. In terms of the subtransfer operator , this means that
For , things are a little more subtle. Nevertheless, it can be shown that
for every fixed real with . Here, as expected, is the symmetric interval . In particular, there is no nontrivial function with for any with and any with .
In , the subtransfer operator was shown to extend to a bounded operator on the space , whose elements are distributions on . The space consists of the restrictions to the open interval of the distributions in the space
supplied with the induced quotient norm, as we mod out with respect to all the distributions whose support is contained in . The quotient norm comes from the norm on the space , which is given by
and we should mention that the is in the natural sense the predual of the real -space on the line, denoted by , which consists of all the functions in whose modified Hilbert transform also is in . In the definition of the space , the letter stands for the Hilbert transform, given by the principal value integral
and is the codimension subspace
By a theorem of Kolmogorov, the Hilbert transform of an is well-defined pointwise almost everywhere as a function in the quasi-Banach space of weak- functions. More generally, if is Lebesgue measurable with positive length, the weak- space consists of all measurable functions with finite quasinorm
where denotes the set
and the absolute value sign assigns the linear length to given set. Kolmogorov’s theorem allows us to think of the distributions (or pseudomeasures) in as elements of , so that in particular, can be identified with a subspace of , the corresponding weak- space on the interval . For the pointwise interpretation, the formula (1.2.5) for the operator remains valid. We will work mainly in the setting of distribution theory. When we need to speak of the pointwise function rather than the distribution , we write in place of , and call it the valeur au point. So “” maps from distributions to functions.
On the space , the subtransfer operators all act contractively. This is not the case with the extension to .
Fix . Then the operator is bounded, but its norm exceeds .
A decomposition analogous to (1.2.1) holds for distributions as well, only we would need two integrals, one with and the other with (and the latter integral should be taken over a bigger interval, e.g. to allow for tails). Thinking physically, we allow for two kinds of “particles”, focused particles as well as spread-out particles . Then is a kind of state space, and acts on this state space. It is then natural to ask whether there is a nontrivial invariant state under . More generally, we would ask whether there exists a with for any scalar with . To appreciate the subtlety of this question, we note that in the slightly larger space , there are plenty of invariant states with , see the example provided in Remark 11.2.1. That example is constructed as the Hilbert transform of the difference of two Dirac point masses, with one point inside and the other point outside . The example in fact suggests that within the space of Hilbert transforms of finite Borel measures, the invariant states might possess an intricate and interesting structure. In the space , which contains the Hilbert transforms of the absolutely continuous measures, this is however not the case.
Fix . For , we have the asymptotic decay in as .
So, although has norm that exceeds on , the orbit of a given converges to in the weaker sense of the quasinorm in . In other words, the -quasinorm serves as a Lyapunov energy for the asymptotic stability of the -orbits. In the setting of the smaller space , this convergence amounts to the statement that the basin of attraction of the attractor contains almost every point of the interval . Apparently, this property extends to the larger space , but not to e.g. (see Remark 11.2.1). The proof of Theorem 1.2.2 is supplied in Subsection 11.2.
Fix . If for some and some scalar with , then .
Our understanding is slightly less complete for . We recall that a distribution, defined on a symmetric interval about , is odd if its action on the even test functions equals .
For odd , we have the asymptotic decay in as for each with .
The proof, which is supplied in Subsection 14.2, is much much more sophisticated than that of Theorem 1.2.2. It uses the full strength of the machinery developed around a subtle dynamical decomposition of the odd part of the Hilbert kernel. We believe that a similar dynamical decomposition is available for the even part of the Hilbert kernel as well, which would remove the need for the oddness assumption. Again, the -quasinorms serve as Lyapunov energy functionals, for each with . In the setting of the smaller space , the corresponding statement is based on the fact that the dynamics of has as a weakly repelling fixed point, so that the ergodic invariant measure for get infinite mass and cannot be in . It follows immediately from Theorem 1.2.4 that the point spectrum of the operator is contained in the open unit disk . In particular, there is no -invariant element of , the subspace of the odd distributions in .
If for some odd and some scalar with , then .
As already mentioned, this corollary is an immediate consequence of Theorem 1.2.4.
From a dynamical perspective, it is quite natural to introduce the odd-even symmetry, as the transformation itself is odd: (except possibly at the endpoints ). E.g., in connection with the partial fraction expansions with even partial quotients, it is standard to keep track of only the orbit of the absolute values on the interval . Note that clearly, the subtransfer operators preserve odd-even symmetry. As for the remaining even symmetry case, we observe that which is even and equals (a constant multiple of) the density of the ergodic invariant measure.
In view of the Observation in Subsection 1.1, Corollaries 1.2.3 and 1.2.5 go beyond the standard notion of ergodicity. The main point is that we insert distribution theory in place of measure theory. We have not been able to find any appropriate references for this in the literature, but suggest some relevance of the works ,  for the discrete setting, and  for flows.
1.3. Applications to the problem of completeness of a system of unimodular functions
As an application to Corollaries 1.2.3 and 1.2.5, we have the following result on the completeness of the nonnegative integer powers of two singular inner functions in the weak-star topology of the space of functions which extend boundedly and holomorphically to the upper half-plane.
Fix two positive reals . Then the functions
which are elements of , span together a weak-star dense subspace of if and only if .
Note that the “only if” part of Theorem 1.3.1, is quite simple, as for instance the work in  shows that in case , the weak-star closure of the linear span in question has infinite codimension in . Hence the main thrust of the theorem is the “if” part. The proof of Theorem 1.3.1 is supplied in two installments: for in Subsection 11.1, and for in Subsection 14.1.
A standard Möbius mapping brings the upper half-plane to the the unit disk , and identifies the space with , the space of all bounded holomorphic functions on . For this reason, Theorem 1.3.1 is equivalent to the following assertion, which we state as a corollary.
Fix two positive reals . Then the linear span of the functions
is weak-star dense in if and only if .
We suppress the trivial proof of the corollary.
Clearly, Theorem 1.3.1 supplies a complete and affirmative answer to Problems 1 and 2 in . We recall the question from : the issue was raised whether the algebra generated by the two inner functions
is weak-star dense set in , without the need to resort to the whole algebra.
The analogue of Theorem 1.3.1 was obtained in . In the context of Theorem 1.3.1, the result leads to completeness in the weak-star topology of , the BMOA space of the upper half-plane. The latter assertion is substantially weaker than Theorem 1.3.1, as it is not difficult to exhibit a sequence of functions in which fails to be weak-star complete in , but is weak-star complete in .
2. Basic properties of the dynamics of Gauss-type maps on intervals
2.1. Notation for intervals
For a positive real , let denote the corresponding symmetric open interval, and let be the positive side of the interval . At times, we will need the half-open intervals and , as well as the closed intervals and .
2.2. Dual action notation
For a Lebesgue measurable subset of the real line , we write
whenever . This will be of interest mainly when is an open interval, and in this case, we use the same notation to describe the dual action of a distribution on a test function. For a set , stands for the characteristic function of , which equals on and vanishes elsewhere. So, in particular, we see that
2.3. Gauss-type maps on intervals
For background material in Ergodic Theory, we refer to e.g. the book .
For , the -step wandering subset is given by
where (-fold composition). We also agree that . The sets get smaller as increases, and we form their intersections
The cone of positive functions consists of all integrable functions with a.e. on the respective interval. Similarly, we say that is positive if a.e. on the given interval.
Fix . Then we have the following assertions:
(i) The operators and are both norm contractions, which preserve the respective cones of positive functions.
(ii) On the positive functions, acts isometrically with respect to the norm.
(iii) If denotes the -step wandering subset given by (2.3.1) above, then for and .
(iv) For , and , we have that as . In particular, as .
(v) For and with mean , we have that as .
(vi) For and , we have that as for each real with .
This is a conglomerate of ingredients from Propositions 3.4.1, 3.10.1, 3.11.3, 3.13.1, 3.13.2, and 3.13.3 in .
2.4. An elementary observation extending the domain of definition for
We begin with the following elementary observation.
Observation. The subtransfer and transfer operators and , initially defined on functions, make sense for wider classes of functions. Indeed, if , then the formulae (1.2.3) and (1.2.5) make sense pointwise, with values in the extended nonnegative reals . More generally, if is complex-valued, we may use the triangle inequality to dominate the convergence of by that of . This entails that is well-defined a.e. if holds a.e. The same goes for of course.
This means that will be well-defined for many functions , not necessarily in .
2.5. Symmetry preservation of the subtransfer operator
The property that preserves symmetry on holds much more generally.
Fix . To the extent that is well-defined pointwise, we have the following:
(i) If is odd, then is odd as well.
(ii) If is even, then is even as well.
This follows from Proposition 3.6.1 in .
Along with the symmetry, we can add constraints like monotonicity and convexity. Under such restraints on , the pointwise values of are guaranteed to exist, and the constraint is preserved under .
Fix . We have the following:
(i) If is odd and (strictly) increasing, then so is .
(ii) If is even and convex, and if , then so is .
This follows from Propositions 3.7.1 and 3.7.2 in .
2.6. Preservation of point values of continuous functions under
For with , let denote the space of continuous functions on the compact symmetric interval .
Fix . If , then . Moreover, if in addition, is odd, then .
This result combines Propositions 3.8.1 and 3.8.2 in .
2.7. Subinvariance of certain key functions
Next, we consider the -iterates of the function
where is assumed confined to the interval . This function is not in , although it is in . However, by the observation made in Subsection 2.4, we may still calculate the expression pointwise wherever . Note that is the invariant measure for the transformation , which in terms of the transfer operator means that .
Fix . For the function , we have that
As for the function , we have the estimate
which for , may be replaced by the uniform estimate
As noted earlier, for , we have the equality .
3. Background material: the Hilbert transform on the line and related spaces
3.1. The Szegő projections and the Hardy -space
For a reference on the basic facts of Hardy spaces and BMO (bounded mean oscillation), we refer to, e.g., the monographs of Duren and Garnett , , as well as those of Stein , , and Stein and Weiss .
Let and be the subspaces of consisting of those functions whose Poisson extensions to the upper half plane
are holomorphic and conjugate-holomorphic, respectively. Here, we use the term conjugate-holomorphic (or anti-holomorphic) to mean that the complex conjugate of the function in question is holomorphic.
It is well-known that any function has vanishing integral,
In other words, , where
By a version of Liouville’s theorem,
which allows us to think of the space
as a linear subspace of . We will call the real -space of the line , although it is -linear and the elements are generally complex-valued. It is not difficult to show that is norm dense as a subspace of . The elements of are just the functions which may be written in the form
As already mentioned, the decomposition (3.1.3) is unique. As for notation, we let and denote the projections and in the decomposition (3.1.3). These Szegő projections can of course be extended beyond this setting; more about this in the following subsection.
3.2. The Hilbert and the modified Hilbert transform
With respect to the dual action
we may identify the dual space of with . Here, is the space of functions of bounded mean oscillation; this is the celebrated Fefferman duality theorem , . As for notation, we write “” to express that we mod out with respect to the constant functions. One of the main results in the theory is the theorem of Fefferman and Stein  which tells us that
or, in words, a function is in if and only if it may be written in the form , where . Here, denotes the modified Hilbert transform, defined for by the formula
The decomposition (3.2.1) is clearly not unique. The non-uniqueness of the decomposition is equal to the intersection space
which we refer to as the real -space.
We should compare the modified Hilbert transform with the standard Hilbert transform , which acts boundedly on for , and maps into for . Here, denotes the weak- space, see e.g. (1.2.9). The Hilbert transform of a function , assumed integrable on the line with respect to the measure , is defined as the principal value integral
If , where , then both and are well-defined a.e., and it is easy to see that the difference equals to a constant. It is often useful to think of the natural harmonic extensions of the Hilbert transforms and to the upper half-plane given by
So, as a matter of normalization, we have that . This tells us the value of the constant mentioned above: .
Returning to the real -space, we note the following characterization of the space in terms of the Hilbert transform: for ,
The Szegő projections and which were mentioned in Subsection 3.1 are more generally defined in terms of the Hilbert transform:
In a similar manner, for , based on the modified Hilbert transform we may define the corresponding projections (which are actually projections modulo the constant functions)
so that, by definition,